Here we are considering:
$\mathbb{S}^n$ as the $n$-sphere
$\mathbb{L}^n$ is the lorentzian $n$-dimensional space, which is just $\mathbb{R}^n$ with the metric given by $\langle u, v \rangle_{\mathbb{L}} = -u_1v_1 + \displaystyle{\sum_{i = 2}^n} u_iv_i$, where $u=(u_1, u_2, \cdots, u_n)$ and $v=(v_1, v_2, ..., v_n) \in \mathbb{R}^n$
$\mathbb{H}^n = \{(x_1, x_2, \cdots, x_{n+1}) \in \mathbb{L}^{n+1} \ : \ -x_1^2 + x_2^2 + \cdots + x_{n+1}^2 = -1, x_1 > 0 \} \subset \mathbb{L}^{n+1}$
$\mathbb{E}^n$ is just $\mathbb{R}^n$ with the usual metric.
$\mathbb{S}^n \times \mathbb{R} = \{(x_1, \cdots, x_{n+2}) \in \mathbb{E}^{n+2} \ : \ x_1^2 + x_2^2 + \cdots + x_{n+1}^2 = 1 \} \subset \mathbb{E}^{n+2}$
$\mathbb{H}^n \times \mathbb{R} = \{(x_1, \cdots, x_{n+2}) \in \mathbb{L}^{n+2} \ : \ -x_1^2 + x_2^2 + \cdots + x_{n+1}^2 = -1 , x_1 > 0\} \subset \mathbb{L}^{n+2}$
I'm interested specifically in $\mathbb{H}^n \times \mathbb{R}$ and $\mathbb{S}^n \times \mathbb{R}$ in the cases $n=1$ and $n=2$. I think $\mathbb{S}^1 \times \mathbb{R}$ is just an ordinary cylinder in $\mathbb{R}^3$ , and I'm aware it's not actually possible to visualize $\mathbb{S}^2 \times \mathbb{R}$ since it's a $4$-dimensional space, but how can one understand it better? I'd really like to understand/(or visualize them as far as such a thing can be done) the spaces $\mathbb{H}^1 \times \mathbb{R}$ and $\mathbb{H}^2 \times \mathbb{R}$. Any help is appreciated.
The trick is to identify $\mathbb{R}$ with $(0,\infty)$, with $-\infty$ to $0$ and $\infty$ to $\infty$. Then though the objects you are interested in are embedded in a 4-space, you can embed them in a 3-space!